Computational schemes for exact linearization of discrete-time systems using a geometric approach

In this work we present a theory for exact linearization of discrete-time systems. Using this theory, a nonlinear system can be transformed to a linear, reachable system through a state coordinate change and a nonlinear state feedback law. This permits the controller design to be carried out in the new coordinates using linear system theory. We assume that the system map f possesses an equilibrium point, and that its Jacobian is full rank around this point. We develop a mathematical framework for the following problems: (i) Exact linearization using state coordinate change, and (ii) Exact linearization using state coordinate change and feedback, also called feedback linearization. Vector fields and one-forms are defined using the tangent map and the co-tangent map induced by the system map. The necessary and sufficient conditions for the first problem are geometric conditions on f related vector fields. The second problem is solved using two different methods. In one method, we define a nested sequence of involutive and constant dimensional distributions using the f-related vector fields. Sufficient conditions for feedback linearizability are expressed in terms of these distributions. Using Frobenius's Theorem we obtain a set of partial differential equations representing the integrability conditions for these distributions. The linearizing transformation and feedback law are constructed from the solution to this set of partial differential equations. In a second approach to the feedback linearization problem, we define column vectors and row-vectors for the discrete-time systems. We show that the necessary and sufficient conditions for feedback linearizability can be rewritten in an easily verifiable form using these column vectors. We define relative degree for the discrete-time system using the row-vectors, and show that the linearizing coordinate transformation and feedback law can be constructed if the nonlinear system has full relative degree. We illustrate our theory through two models of electrically stimulated muscle from biomedical control engineering.